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Properties

Label 2008.4.a.c
Level $2008$
Weight $4$
Character orbit 2008.a
Self dual yes
Analytic conductor $118.476$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2008,4,Mod(1,2008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2008.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2008 = 2^{3} \cdot 251 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2008.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(118.475835292\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 11 q^{3} - 31 q^{5} - 71 q^{7} + 489 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 11 q^{3} - 31 q^{5} - 71 q^{7} + 489 q^{9} - 132 q^{11} - 93 q^{13} - 251 q^{15} - 90 q^{17} - 367 q^{19} - 99 q^{21} - 562 q^{23} + 1559 q^{25} - 461 q^{27} - 364 q^{29} - 552 q^{31} - 78 q^{33} - 882 q^{35} - 199 q^{37} - 1079 q^{39} + 192 q^{41} - 1197 q^{43} - 1324 q^{45} - 1685 q^{47} + 3167 q^{49} - 895 q^{51} - 1424 q^{53} - 1130 q^{55} - 273 q^{57} - 1823 q^{59} - 726 q^{61} - 2547 q^{63} - 291 q^{65} - 2780 q^{67} - 854 q^{69} - 3063 q^{71} + 176 q^{73} - 1756 q^{75} - 1858 q^{77} - 1993 q^{79} + 5698 q^{81} - 2854 q^{83} - 1570 q^{85} - 1638 q^{87} - 59 q^{89} - 2160 q^{91} - 3956 q^{93} - 4305 q^{95} - 756 q^{97} - 5017 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −10.0846 0 −5.89682 0 −15.9708 0 74.6997 0
1.2 0 −9.57494 0 9.33217 0 9.70777 0 64.6794 0
1.3 0 −9.54342 0 −4.49657 0 15.0683 0 64.0770 0
1.4 0 −9.42753 0 13.1979 0 10.2624 0 61.8784 0
1.5 0 −9.21311 0 −18.3228 0 29.9868 0 57.8813 0
1.6 0 −8.87446 0 18.9783 0 −28.1514 0 51.7560 0
1.7 0 −8.24401 0 −1.55575 0 −34.0369 0 40.9638 0
1.8 0 −7.41853 0 20.3771 0 −19.4363 0 28.0346 0
1.9 0 −6.82298 0 −6.53711 0 25.6622 0 19.5530 0
1.10 0 −6.36012 0 −4.50955 0 33.5053 0 13.4511 0
1.11 0 −5.72839 0 2.64954 0 −7.10776 0 5.81447 0
1.12 0 −5.67915 0 −18.9564 0 −22.1236 0 5.25275 0
1.13 0 −5.21628 0 8.24798 0 −5.99529 0 0.209617 0
1.14 0 −5.14880 0 −17.2771 0 −32.9456 0 −0.489833 0
1.15 0 −4.81107 0 −12.5553 0 7.59971 0 −3.85362 0
1.16 0 −4.13499 0 −12.0479 0 2.03666 0 −9.90185 0
1.17 0 −3.99286 0 10.3257 0 −21.1888 0 −11.0570 0
1.18 0 −3.56612 0 −9.75482 0 −29.2365 0 −14.2828 0
1.19 0 −3.12416 0 16.2839 0 22.5023 0 −17.2396 0
1.20 0 −3.03774 0 15.0797 0 11.9749 0 −17.7721 0
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(251\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2008.4.a.c ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.4.a.c ✓ 50 1.a even 1 1 trivial